Role of chemical disorder in tuning the Weyl points in vanadium doped Co_2TiSn
Payal Chaudhary, Ajit K. Jena, Krishna Kant Dubey, Gaurav K. Shukla, Sudipta Kanungo, S.-C Lee, S. Bhattacharjee, Jan Minár, Sunil Wilfred D'Souza, Sanjay Singh
RRole of chemical disorder in tuning the Weyl points in vanadium doped Co TiSn
Payal Chaudhary, Krishna Kant Dubey, Gaurav K. Shukla, and Sanjay Singh
School of Materials Science and Technology, Indian Institute of Technology (Banaras Hindu University), Varanasi 221005, India
Sudipta Kanungo
School of Physical Sciences, Indian Institute of Technology Goa, Goa 403401, India
Ajit K. Jena, S.-C Lee, and S. Bhattacharjee
Indo-Korea Science and Technology Center (IKST), Bangalore 560065, India
Jan Min´ar and Sunil Wilfred D’Souza,
New Technologies Research Centre, University of West Bohemia,Univerzitn´ı 8, CZ-306 14 Pilsen, Czech Republic
The lack of time-reversal symmetry and Weyl fermions give exotic transport properties to Co-based Heusler alloys. In the present study, we have investigated the role of chemical disorder on thevariation of Weyl points in Co Ti V x Sn magnetic Weyl semimetal candidate. We employ the firstprinciple approach to track the evolution of the nodal lines responsible for the appearance of Weylnode in Co TiSn as a function of V substitution in place of Ti. By increasing the V concentrationin place of Ti, the nodal line moves toward fermi level and remains at Fermi level around the middlecomposition. Further increase of the V content, leads shifting of nodal line away from Fermi level.Density of state calculation shows half-metallic behavior for the entire range of composition. Themagnetic moment on each Co atom as a function of V concentration increases linearly up to x=0.4,and after that, it starts decreasing. The first-principles calculations reveal that via replacing almosthalf of the Ti with V, the intrinsic anomalous Hall conductivity increased twice as compared to theundoped composition. Our results indicate that the composition close to the 50% V doped Co TiSn,will be an ideal composition for the experimental investigation of Weyl physics.
I. INTRODUCTION
Weyl semimetals (WSMs) have created vast interest inrecent years due to their novel electronic and transportproperties [1, 2], such as very high electron mobilities [3],Fermi arcs on the surface [4, 5], extremely large magne-toresistance [3, 6], anomalous Hall effect [7, 8], and theanomalous Nernst effect [9, 10]. WSMs also exhibit un-conventional optical properties, such as large and quan-tized photocurrents [11–14], second-harmonic generation[15, 16], and Kerr rotation [17, 18]. These properties canlead to more efficient electronic and photonic applications[1]. WSMs are an especial class of the topological mate-rials, characterized by the crossings of singly degenerateenergy bands near the Fermi energy, leading to the for-mation of pairs of Weyl nodes [19–21]. WSMs provide aplatform for manipulating and understanding the physicsof the chiral Weyl fermions [22, 23].Inversion symmetry (IS) or time-reversal symmetry(TRS) must be broken to obtain Weyl nodes/WSMs[2, 23, 24]. WSMs with broken IS have been investigatedextensively [3, 19, 20], while WSMs with broken TRS,known as magnetic WSMs, were recently discovered inexperiments [9, 25]. Magnetic WSMs created much in-terest because, in this class of WSMs, the properties canbe manipulated using a magnetic field as an external de-gree of freedom. Heusler alloys have emerged as an im-portant class of materials to investigate the Weyl physicsand its consequences [8, 25–31]. In Heusler compounds,we either look at half-Heusler or inverse Heusler com- pounds (IS breaking) [26], at magnetic compounds (TRSbreaking) [25, 27–29], or compounds with both IS andTRS breaking [8, 30, 31]. In most magnetic Heuslers,the magnetization direction can be changed quite easily.Since the location of Weyl nodes in the momentum spacedepends on the direction of magnetization, Heusler com-pounds can prove to be useful to understand the physicsof Weyl fermions. Combined with their extensive tunabil-ity, Heusler WSMs are a promising platform for practicaltopological applications [1, 2].Co MnGa has been theoretically predicted and ex-perimentally proven to be a WSM. The Weyl nodeslie close to the Fermi energy [25, 32, 33], and trans-port measurements have shown large anomalous Halland Nernst effects [10, 34–38]. Besides Co MnGa, otherHeusler compounds have also been predicted to be WSM[39–42]. Although, for most of the proposed HeuslerWSMs, the Weyl nodes lie away from the Fermi energy,which reduces the topological properties of these mate-rials [27, 28, 39–42]. By tuning the Fermi energy, it cancoincide with the energy of the nodes, which can signifi-cantly improve the properties [41, 43, 44].Co TiSn, a Heusler compound that has a high Curietemperature and shows half-metallic behavior [45–49],has been proposed as a WSM candidate [28, 40, 41].Co TiSn has 26 valence electrons and has Weyl nodesa few hundred meV above the Fermi energy [28, 40, 41].The number of valence electrons must be increased tomake the nodes’ energy coincide with the Fermi energy[28]. To achieve this, Ti, which has 2 electrons in its va- a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b lence 3 d orbital, can be substituted with V, which has 3electrons in the 3 d orbital. Wang et al. suggested thatthe doping of 10% V in place of Ti (i.e., Co Ti . V . Ga)to obtain the WSM phase [41]. However, efforts to syn-thesize thin films Co Ti . V . Sn have not been success-ful, and the most stable composition was found to beCo Ti . V . Sn (i.e., 40% V doped) [50]. Transport mea-surements for Co TiSn and Co Ti V Sn thin filmsshow anomalous Hall and Nernst effects [50, 51]. Inter-estingly, the anomalous Nernst angle and coefficient forthe doped compound (i.e., Co Ti . V . Sn) were signifi-cantly higher than the undoped one [51].In this manuscript, we investigated the effect of V dop-ing on the electronic structure of Co Ti V x Sn (x = 0.0,0.2, 0.4, 0.6, 0.8 and 1.0). The nodal lines responsible forWeyl node formation in Co TiSn, changes as a function ofV substitution in place of Ti. With increasing V in placeof Ti nodal line get shifted towards the Fermi level closeto the mid composition (i.e Co Ti . V . Sn).The intrin-sic anomalous Hall conductivity obtained from the the-ory for the 50% V doped composition (Co Ti . V . Sn)is nearly twice as compared to the undoped composition.
II. METHODS
The calculations were performed using the full-potential Korringa–Kohn–Rostoker (KKR) Green func-tion method, as implemented in the SPRKKR package[54]. The exchange-correlation energies were describedusing the generalized gradient approximation (GGA)with PBE parameterization [55]. A k-mesh consistingof (22 × ×
22) k-points was used, and the angular mo-mentum cut-off number was chosen to be l max = 3. TheFermi energies were determined using the Lloyd formula[56]. The calculations for the Bloch spectral functionstook spin-orbit coupling (SOC) into account. The bandstructures were calculated for both the spin-polarized,non spin-orbit coupling case as well as the case withspin-orbit coupling. The ground state energies obtainedwhen the magnetization was kept in the [001], [110] and[111] directions are found to be the same within the lim-its of the software, indicating that the magnetization di-rections, and therefore the positions of the Weyl nodes,can be changed easily. This matches with the literatureon Co TiSn, Co VSn and similar Heusler compounds[40, 41]. The disorder was taken into account throughthe coherent potential approximation (CPA) [57, 58].The full electron and Wannier interpolated bands aswell as the anomalous Hall conductivity (AHC) was cal-culated using pseudo-potential based density-functionaltheory (DFT) and wannier90 as implemented in Quan-tum ESPRESSO (QE) [59–62]. The exchange-correlationpotential is approximated through PBE-GGA functional[55]. Optimized norm-conserving Vanderbilt pseudopo-tentials [63] are used in the calculations and the kineticenergy cutoff for the planewave is taken as 80 Ry . Theelectronic integration over the Brillouin zone is approx- imated by the Gaussian smearing of 0.005 Ry both forthe self-consistent (sc) and non-self-consistent (nsc) cal-culations. The Monkhorst-Pack k -grid of 8 × × d -orbitals are used as the projections forthe wannier90 calculations. The AHC calculation is car-ried out with a dense k -grid of 75 × ×
75. Further,through the adaptive refinement technique a fine meshof 5 × × | Ω( k ) | ) exceeds 100 Bohr . Allthe calculated structures are optimized with tight con-vergence threshold both for the energy (10 − Ry ) andforce (10 − Ry/Bohr ).The threshold for self-consistentenergy calculations is taken as 10 − Ry . III. RESULTS AND DISCUSSIONIII.1. Optimisation of lattice parameters
The lattice parameters for all the compositions wereobtained by varying the parameters and calculating therespective ground state energies, using SPRKKR soft-ware. The equilibrium values were found using the Birch-Murnaghan equation of state [64] fit for the total en-ergy as a function of the unit cell volume. The plotsof total energy vs. the unit cell volume are shown inFig. 1. The calculated lattice parameters values, 6.104˚A(for x=0.0), 6.087 ˚A(for x=0.2), 6.083 ˚A(for x=0.4),6.065 ˚A(for x=0.6), 6.050 ˚A(for x=0.8) and 6.04 ˚A(forx=1.0) are in well agreement with the experimentallyreported lattice parameters 6.076˚A, 6.051 ˚A, 6.040 ˚A,6.034˚A, 6.014˚A, 5.98 ˚A for x = 0.0, 0.2, 0.4, 0.6, 0.8, and1.0 respectively, which follow similar trend with changein the composition[52, 53].
III.2. Band structure calculations forstoichiometric Co TiSn and Co VSn Heuslercompounds
The band structures for the end (stoichiometric) com-positions, with spin-orbit coupling (SOC), are shown inFig. 2. The band structures for both the stoichiometriccompositions, Co TiSn and Co VSn, have been calcu-lated using the Heusler (L2 ) cubic structure with spacegroup F m ¯3 m . In this structure, Co occupies the 8c (1/4,1/4, 1/4) Wyckoff position, Ti (or V) occupies 4b (1/2,1/2, 1/2), and Sn is at 4a (0, 0, 0), as shown in Fig. 2(a)for Co TiSn as an example.Fig. 2 shows the spin-orbit coupled band structures forCo TiSn and Co VSn with magnetization oriented alongthe [110] and [001] directions. Red circles mark the nodallines of interest which form the Weyl nodes. An analysisof the hybridisation and symmetry of these nodal lines isdetailed in the appendix. As illustrated in Fig. 2(b), theband structures cover the high-symmetry lines in the xy -plane of the Brillouin zone (BZ). In Co TiSn, the nodalFIG. 1: Total energy vs. volume of the unit cell forCo Ti − x V x Sn (x = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0). TheBirch-Murnaghan equation of state is used to determinethe equilibrium values [64].line remains unaffected when the magnetization is ori-ented in the [001] direction (Fig. 2(d)), but very slightgaps occur when it is oriented in the [110] direction (Fig.2(c)). In the band structure for the [110] magnetizationdirection (Fig. 2(c)), the nodal line crossing along Γ − W has a very small gap, not visible in the figure, while thecrossings along Γ − X and Γ − K remain unaffected. Thecrossings of the nodal line bands with the surroundingband along Γ − W and Γ − K also have gaps, accordingto the magnetization direction, as can be seen next tothe red circles in the figures (Fig. 2(c) and 2(d)). InCo VSn, these nodal lines lie entirely below the Fermienergy, as seen in Fig. 2(e) and 2(f). FIG. 2: (a)Structure of Co TiSn. Co atoms arerepresented by green spheres, Ti atoms by blue spheres,and Sn atoms by yellow spheres. (b) Brillouin zoneshowing the high-symmetry points and the k-pathfollowed in the band structures. (c),(e) Band structuresfor [110] magnetization with spin-orbit coupling.(d),(f)Band structures for [001] magnetization withspin-orbit coupling. The red circles indicate the locationof the crossings in the Co-Y (Y = Ti, V) hybridized 3dbands.
III.3. Bloch spectral functions for Co Ti V x Sn For disordered compounds, it is difficult to determinethe E versus k dispersion relations using methods forperiodic ordered systems, such as DFT. One approach isto construct a supercell to add the substituted elementin the required ratio and calculate the dispersion rela-tions.However, this results in complex band structureswith additional bands due to symmetry. These super-cell bands can then be unfolded to get an effective bandstructure [67, 68]. Using this approach to calculate therelations for a range of compositions is cumbersome andis feasible only for specific compositions.Here, we use Bloch spectral functions to representthe electronic structure. The Bloch spectral function A B ( k , E ), defined as the Fourier transform of the Green’sfunction G ( r , r (cid:48) , E ), can be written asFIG. 3: Bloch spectral function plots of Co Ti V x Sn. The majority and minority spin states are represented bythe blue and red lines, respectively. The band crossing points can be seen shifting downwards with respect to theFermi level as the concentration of V increases, tuning with the Fermi level at x = 0.2 and x = 0.4. A B ( k , E ) = − πN Im T r { N (cid:88) n,n (cid:48) e i k ( R n − R n (cid:48) ) × (cid:90) Ω d rG ( r + R n , r + R n (cid:48) , E ) } (1)This function can be interpreted as the k -resolved den-sity of states [56].The Bloch spectral function (BSF) plots are shown inFig. 3.The nodal line can be seen shifting downwardwith respect to the Fermi energy as the concentration ofV increases. At x = 0.4, the point of highest energy ofthe nodal line tunes with the Fermi energy, along Γ − K .This has been also seen in the Fermi surface plots (Fig. 4)where for x = 0 .
4, the two bands touch at the extremities,along Γ − K . For higher V doped composition e.g. x= 0.6,the nodal lines lie entirely below the Fermi energy.Thismeans that for all compositions having V concentrationin the range of 0 . ≤ x ≤
1, the nodal lines lie entirelybelow the Fermi level.A distinct feature of the BSF plots of the substitutedcompounds are the broadening of the majority bands.This broadening occurs in the energy range -0.5 eV to2.0 eV. These are the same Co-X (X = Ti, V) hybridized3 d bands which form the nodal lines of interest. In thesubstituted compounds, the bands in this energy rangeare primarily formed by the 3 d states on the 4b Wyckoff position, which contain the Ti and V atoms, and havea negligible contribution from Co atoms. The Co atomsin the substituted compounds have more contribution inthe states below the nodal lines. The broadening in thebands occurs due to the randomly substituted additional3 d electron of the V atom. Since the band correspondingto the additional electron has a different energy, randomfluctuations are induced in the energy window of the 3 d states. The new electronic state manifests as an inter-mediate state, which can be seen becoming more welldefined as the concentration of V increases. III.4. Fermi surface
The evolution of the Fermi surface with respect to theV concentration can be seen in Fig. 4. The Fermi surfaceplots show the electronic states in the xy -plane of the BZ,at k z = 0, lying on the Fermi energy. On the right is aschematic diagram showing the high-symmetry points inthe Brillouin zone and a cross-section of the Brillouinzone at k z = 0. The intermediate state formed by theaddition of the 3 d V electron can be seen becoming morewell-defined along the X − Γ and W − K directions asthe concentration of V increases. The bands which formthe nodal lines can be seen clearly in the Fermi surfaceplots; one has a Fermi surface around the Γ point andthe other around the K point. At x = 0 .
2, the nodal linetunes with the Fermi energy along Γ − W , as can be seenFIG. 4: Fermi surface plots of Co Ti V x Sn. Majority spin states are presented in blue. Since all the compositionsare half-metallic, there are no minority spin states at the Fermi level. On the right is a diagram of the Brillouin zone(BZ) showing the high-symmetry lines and points, along with the Fermi surface cross-section (cut through the BZ at k z = 0). The green region shows the first BZ.in both the BSF and Fermi surface plots.At x = 0 .
4, theFermi surface plot shows the two bands touching at theextremities, along Γ − K . For 0 . ≤ x ≤
1, the nodal lineslie entirely below the Fermi energy.
III.5. Density of states and magnetization
The stoichiometric compounds Co TiSn and Co VSnhave been reported as half-metallic ferromagnets [45, 69–72]. To investigate the half-metallic character as a func-tion of chemical disorder, we performed density of states(DOS) calculations as a function of x in Co Ti V x Sn.The evolution of the density of states (DOS) is given inFig. 5. The Fermi energy always lies in the minority spinband gap for all compositions, indicating half-metallic be-havior throughout. The magnitude of the gap increasesslightly with the increase in V concentration, and the gapfor the stoichiometric compounds is in good agreementwith the literature. The band gap (∆E) for compositionsx=0.0, 0.2, 0.4, 0.6, 0.8 and 1.0 is found to be 0.491 eV,0.505 eV, 0.518 eV, 0.518 eV, 0.532 eV and 0.546 eV, re-spectively. This reveals that the ∆E is slightly increasingwith increasing V substitution in place of Ti.Having obtained the evaluation of nodal lines and half-metallic character, we turn our discussion regarding themagnetic behavior of Co Ti V x Sn alloys. The mag-netic moments calculated for all the compounds are given in Table I. Interestingly the magnetic moment of Co in-creases initially and after that, it starts decreasing, whichis in very well agreement with the experimental findings(Table I) [52, 53]. The magnetic moment of Co is orientedantiparallel with the Ti moment and parallel with the Vmoment. Ti has a low magnetic moment of around 0.1 µ B per atom, while V has a higher moment ranging from0.73 to 0.88 µ B per atom. The total magnetic momentper formula unit obtained here shows an increasing trendwith V substitution and found 3 µ B , which is well agree-ment with previous reported literature[75]. Thus our allresults are in accordance with the observed behavior inthe experiments and explain the unusual large anomalousNernst coefficient for the Co Ti . V . Sn as compared tothe stoichiometric compound Co TiSn [51].
III.6. Anomalous Hall conductivity
The anomalous Hall effect/conductivity is direct con-sequence of the Berry curvature of the electronic bandstructure near the Fermi level, which act as pseudo-magnetic field in momentum space.The Berry curvaturenear the Fermi level introduces a transverse momentumin electron motion and derives large anomalous Hall con-ductivity(AHC). So in our case the middle composition isinterest of research due to fact that Weyl nodes situatednear the Fermi level.To substantiate our compositionalFIG. 5: Density of states of Co Ti V x Sn (x=0.0, 0.2, 0.4, 0.6, 0.8 and 1.0). For each compound, the positive y-axisrepresents the density of the majority spin states, and the negative y-axis represents the minority spin states.Experimental ( µ B ) Calculated ( µ B ) x µ Co [53] µ Co µ T i µ V µ total . . . . . . Ti V x Sn (x=0.0,0.2, 0.4, 0.6, 0.8 and 1.0). The moments for individualatoms are given in units of µ B per atom, and the totalmoment is given in µ B per formula unit.dependent theoretical analysis on variation of position ofWeyl point in band structure, we theoretically calculatedthe AHC with the expectation that large AHC shouldexhibit around the mid composition. To calculate the in-trinsic anomalous Hall conductivity (AHC) for the pureand doped systems, the conventional unit cells have beenconsidered (unless otherwise specified). The Bloch wavefunctions are projected into maximally localized Wan-nier functions in order to compute the intrinsic AHC.The intrinsic AHC is proportional to the Brillouin zone(BZ) summation of the Berry curvature over all occupiedstates [62, 73, 76] σ xy = − e (cid:126) (cid:88) n (cid:90) BZ d k (2 π ) Ω n,z ( k ) f n ( k ) , (2)where f n ( k ) is the Fermi distribution function for theband n , Ω n,z ( k ) is the z component of the Berry curva-ture at the wave vector k . The Berry curvature is relatedto the Berry connection ( A n ( k )) as Ω n ( k ) = ∇ k × A n ( k ) , (3)where ” n ” is the band index and A n ( k ) in terms ofcell-periodic Bloch states | u n k (cid:105) = e − i k.r | ψ n k (cid:105) is definedas A n ( k ) = (cid:104) u n k | i ∇ k | u n k (cid:105) [62].Co TiSn possesses the fcc L (space group m x , m y and m z in the absence of any net magnetic moment. These mir-ror planes protect the gapless nodal lines in the bandstructure in k x = 0, k y = 0, and k z = 0 planes [40, 41].To compute AHC, the spin-orbit coupling is introducedand the direction of the magnetization has been set along(001). Therefore, due to symmetry breaking of mirrorplanes the nodal lines in the k x = 0 and k y = 0 planeswill exhibit a finite band gap while the gapless nodal linewill survive only along the magnetization direction (inthe k z = 0 plane) [40, 41]. However, due to helical distri-bution of Berry curvature around this gapless nodal line,in the mirror plane, the total flux is zero and therebyit does not contribute to the intrinsic AHC [28]. Con-currently, the Berry curvature around the broken nodallines is oriented along the direction of magnetization andcontributes to the intrinsic AHC [28]. Our calculated in-trinsic AHC value (99.38 S/cm and 92.40
S/cm for theprimitive cell) at E F for the pure system is in excellentagreement with the value (100 S/cm ) reported in litera-ture [28]. In addition to this, as it is shown in Fig. 6 theintrinsic anomalous Hall conductivity is almost constantin the vicinity of E F .Earlier, we have seen that the band crossing points ofthe pure compound (Co TiSn) shift downwards with re-spect to the Fermi level ( E F ) as the concentration of V increases (Fig. 2), and particularly, two of such crossingpoints which lie in the unoccupied region (for the sto-ichiometric composition Co TiSn) come close to E F ascomposition approaches toward 50% V doped Co TiSn.FIG. 6: (a) Comparison of Wannier interpolated bandstructure (red) with the full electronic band structure(blue) of Co TiSn. The Fermi energy is set to 0 eV . (b)The calculated intrinsic anomalous Hall conductivity atdifferent energies. The conductivity is found to beconstant in the vicinity of E F ( E F ± eV ).Hence, it is expected that the intrinsic AHC will be en-hanced for the 50% V doped Co TiSn compositions dueto strong energy dispersion.Considering that the calcula-tion with x= 0.4 doping concentration is computationallyvery expensive, we have chosen x= 0.5 to calculate in-trinsic AHC, which is expected to provide us maximumvalue. While simulating AHC as a function of energy,around 0.25 eV above the E F , we get ∼ E F . The same has beenfurther confirmed when the bands are filled along the+ ve energy (w.r.t E F ), which is achieved in the form ofV doping. The calculated AHC along the direction ofmagnetization at E F in 50 % V doped Co TiSn is foundto be 196.84
S/cm , nearly twice of the AHC in the pure system(Co TiSn). Hence, the higher AHC value (in 50% doped system as well as at ∼ eV above the E F in the pure compound) is attributed to the presence ofnodal lines that are very close to E F . IV. CONCLUSION
To summarise, we performed ab initio calculations onthe Co-based Heusler compounds Co Ti V x Sn with x= 0.0, 0.2, 0.4, 0.6, 0.8 and 1.0. We have calculated theband structures, Bloch spectral functions and DOS usingKKR-GF methods. We have also calculated the intrinsicAHC for the x = 0.0, 0.5 and 1.0 compositions. We foundthat nodal lines shift with V substitution and the pointof highest energy of the nodal line responsible for Weylnodes tunes with the Fermi energy for Co Ti V Sn.For composition between x= 0.6 and 1 the nodel line lieentirely below the Fermi energy. We observed a half-metallic character for the entire range of composition.The magnetic moment on each Co atom as a function ofV concentration increases linearly up to x=0.4 and there-after, it starts decreasing. The intrinsic AHC was foundto increase by nearly twice for the 50% doped systemas compared to the undoped composition. Our studysuggests that Co Ti V x Sn series of Heusler alloys ingeneral and Co Ti V Sn composition in particular isimportant to investigate Weyl physics and various exotictransport phenomena.
V. ACKNOWLEDGMENTS
SS thanks Science and Engineering Research Boardof India for financial support through the awardof Ramanujan Fellowship (grant no: SB/S2IRJN-015/2017), Early Career Research Award (grant no:ECR/2017/003186). S.W.D and J.M would like to thankCEDAMNF project financed by the Ministry of Educa-tion, Youth and Sports of Czech Republic, Project No.CZ.02.1.01/0.0/0.0/15.003/0000358 and also for the sup-port by the GA ˇCR via the project 20-18725S. [1] J. Hu, S.-Y. Xu, N. Ni, and Z. Mao, Transport of topo-logical semimetals, Annual Review of Materials Research (2019).[2] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyland dirac semimetals in three-dimensional solids, Rev.Mod. Phys. , 015001 (2018).[3] C. Shekhar, A. K. Nayak, Y. Sun, M. Schmidt, M. Nick-las, I. Leermakers, U. Zeitler, Y. Skourski, J. Wosnitza,Z. Liu, et al. , Extremely large magnetoresistance and ul-trahigh mobility in the topological weyl semimetal can-didate nbp, Nature Physics , 645 (2015). [4] X. Wan, A. M. Turner, A. Vishwanath, and S. Y.Savrasov, Topological semimetal and Fermi-arc surfacestates in the electronic structure of pyrochlore iridates,Physical Review B , 205101 (2011).[5] S. Jia, S.-Y. Xu, and M. Z. Hasan, Weyl semimetals,Fermi arcs and chiral anomalies, Nature materials ,1140 (2016).[6] D. Son and B. Spivak, Chiral anomaly and classical neg-ative magnetoresistance of weyl metals, Physical ReviewB , 104412 (2013).[7] A. Burkov, Anomalous hall effect in weyl metals, Physicalreview letters , 187202 (2014). [8] C. Shekhar, N. Kumar, V. Grinenko, S. Singh, R. Sarkar,H. Luetkens, S.-C. Wu, Y. Zhang, A. C. Komarek,E. Kampert, et al. , Anomalous hall effect in weylsemimetal half-heusler compounds rptbi (r= gd and nd),Proceedings of the National Academy of Sciences ,9140 (2018).[9] M. Ikhlas, T. Tomita, T. Koretsune, M.-T. Suzuki,D. Nishio-Hamane, R. Arita, Y. Otani, and S. Nakatsuji,Large anomalous nernst effect at room temperature in achiral antiferromagnet, Nature Physics , 1085 (2017).[10] A. Sakai, Y. P. Mizuta, A. A. Nugroho, R. Sihomb-ing, T. Koretsune, M.-T. Suzuki, N. Takemori, R. Ishii,D. Nishio-Hamane, R. Arita, et al. , Giant anomalousnernst effect and quantum-critical scaling in a ferromag-netic semimetal, Nature Physics , 1119 (2018).[11] K. Taguchi, T. Imaeda, M. Sato, and Y. Tanaka, Photo-voltaic chiral magnetic effect in weyl semimetals, PhysicalReview B , 201202 (2016).[12] C.-K. Chan, N. H. Lindner, G. Refael, and P. A. Lee,Photocurrents in weyl semimetals, Physical Review B ,041104 (2017).[13] F. de Juan, A. G. Grushin, T. Morimoto, and J. E.Moore, Quantized circular photogalvanic effect in weylsemimetals, Nature communications , 15995 (2017).[14] G. B. Osterhoudt, L. K. Diebel, X. Yang, J. Stanco,X. Huang, B. Shen, N. Ni, P. Moll, Y. Ran, and K. S.Burch, Colossal photovoltaic effect driven by the singu-lar berry curvature in a weyl semimetal, arXiv preprintarXiv:1712.04951 (2017).[15] T. Morimoto and N. Nagaosa, Topological nature ofnonlinear optical effects in solids, Science advances ,e1501524 (2016).[16] L. Wu, S. Patankar, T. Morimoto, N. L. Nair, E. Thewalt,A. Little, J. G. Analytis, J. E. Moore, and J. Orenstein,Giant anisotropic nonlinear optical response in transitionmetal monopnictide weyl semimetals, Nature Physics ,350 (2017).[17] W. Feng, G.-Y. Guo, J. Zhou, Y. Yao, and Q. Niu, Largemagneto-optical kerr effect in noncollinear antiferromag-nets mn 3 x (x= rh, ir, pt), Physical Review B , 144426(2015).[18] T. Higo, H. Man, D. B. Gopman, L. Wu, T. Koretsune,O. M. van’t Erve, Y. P. Kabanov, D. Rees, Y. Li, M.-T. Suzuki, et al. , Large magneto-optical kerr effect andimaging of magnetic octupole domains in an antiferro-magnetic metal, Nature photonics , 73 (2018).[19] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neu-pane, G. Bian, C. Zhang, R. Sankar, G. Chang,Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng,J. Ma, D. S. Sanchez, B. Wang, A. Bansil,F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z.Hasan, Discovery of a weyl Fermion semimetal andtopological Fermi arcs, Science , 613 (2015),https://science.sciencemag.org/content/349/6248/613.full.pdf.[20] S.-Y. Xu, N. Alidoust, I. Belopolski, Z. Yuan, G. Bian,T.-R. Chang, H. Zheng, V. N. Strocov, D. S. Sanchez,G. Chang, C. Zhang, D. Mou, Y. Wu, L. Huang, C.-C. Lee, S.-M. Huang, B. Wang, A. Bansil, H.-T. Jeng,T. Neupert, A. Kaminski, H. Lin, S. Jia, and M. Za-hid Hasan, Discovery of a weyl Fermion state with Fermiarcs in niobium arsenide, Nature Physics , 748 (2015).[21] M. Z. Hasan, S.-Y. Xu, I. Belopolski, and S.-M. Huang,Discovery of weyl Fermion semimetals and topologicalFermi arc states, Annual Review of Condensed Matter Physics , 289 (2017), https://doi.org/10.1146/annurev-conmatphys-031016-025225.[22] X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang,H. Liang, M. Xue, H. Weng, Z. Fang, et al. , Observa-tion of the chiral-anomaly-induced negative magnetore-sistance in 3d weyl semimetal taas, Physical Review X ,031023 (2015).[23] B. Yan and C. Felser, Topological materials: Weylsemimetals, Annual Review of Condensed Matter Physics , 337 (2017).[24] S. Kar and A. M. Jayannavar, Weyl semimetals: Downthe discovery of topological phases, arXiv preprintarXiv:1902.01620 (2019).[25] I. Belopolski, D. S. Sanchez, G. Chang, K. Manna,B. Ernst, S.-Y. Xu, S. S. Zhang, H. Zheng, J. Yin,B. Singh, et al. , A three-dimensional magnetic topologi-cal phase, arXiv preprint arXiv:1712.09992 (2017).[26] X. Liu, L. Li, Y. Cui, J. Deng, and X. Tao, A nonmag-netic topological weyl semimetal in quaternary heuslercompound craltiv, Applied Physics Letters , 122104(2017).[27] K. Manna, L. Muechler, T.-H. Kao, R. Stin-shoff, Y. Zhang, J. Gooth, N. Kumar, G. Kreiner,K. Koepernik, R. Car, et al. , From colossal to zero: con-trolling the anomalous hall effect in magnetic heuslercompounds via berry curvature design, Physical ReviewX , 041045 (2018).[28] B. Ernst, R. Sahoo, Y. Sun, J. Nayak, L. M¨uchler, A. K.Nayak, N. Kumar, J. Gayles, A. Markou, G. H. Fecher, et al. , Anomalous hall effect and the role of berry curva-ture in co 2 tisn heusler films, Physical Review B ,054445 (2019).[29] R. P. Dulal, B. R. Dahal, A. Forbes, N. Bhattarai, I. L.Pegg, and J. Philip, Weak localization and small anoma-lous hall conductivity in ferromagnetic weyl semimetalco 2 tige, Scientific reports , 3342 (2019).[30] Y. Nakajima, R. Hu, K. Kirshenbaum, A. Hughes,P. Syers, X. Wang, K. Wang, R. Wang, S. R. Saha,D. Pratt, et al. , Topological rpdbi half-heusler semimet-als: A new family of noncentrosymmetric magnetic su-perconductors, Science advances , e1500242 (2015).[31] H. Kim, K. Wang, Y. Nakajima, R. Hu, S. Ziemak,P. Syers, L. Wang, H. Hodovanets, J. D. Denlinger, P. M.Brydon, et al. , Beyond triplet: Unconventional supercon-ductivity in a spin-3/2 topological semimetal, Science ad-vances , eaao4513 (2018).[32] D. Multer, G. Chang, S. Xu, X. Zhou, S.-M. Huang,B. Singh, B. Wang, I. Belopolski, J. Yin, S. S. Zhang, et al. , Topological hopf and chain link semimetal statesand their application to co2mnga, in APS Meeting Ab-stracts (2018).[33] I. Belopolski, K. Manna, D. S. Sanchez, G. Chang,B. Ernst, J. Yin, S. S. Zhang, T. Cochran, N. Shumiya,H. Zheng, et al. , Discovery of topological weyl Fermionlines and drumhead surface states in a room temperaturemagnet, Science , 1278 (2019).[34] H. Reichlova, R. Schlitz, S. Beckert, P. Swekis,A. Markou, Y.-C. Chen, D. Kriegner, S. Fabretti,G. Hyeon Park, A. Niemann, et al. , Large anomalousnernst effect in thin films of the weyl semimetal co2mnga,Applied Physics Letters , 212405 (2018).[35] S. N. Guin, K. Manna, J. Noky, S. J. Watzman, C. Fu,N. Kumar, W. Schnelle, C. Shekhar, Y. Sun, J. Gooth, et al. , Anomalous nernst effect beyond the magnetization scaling relation in the ferromagnetic heusler compound co2 mnga, NPG Asia Materials , 16 (2019).[36] S. Takashi, S. Kokado, M. TSUJIKAWA, T. OGAWA,S. Kosaka, M. Shirai, and M. TSUNODA, Signs ofanisotropic magnetoresistance in co 2 mnga heusler alloyepitaxial thin films based on current direction, AppliedPhysics Express (2019).[37] A. Markou, D. Kriegner, J. Gayles, L. Zhang, Y.-C.Chen, B. Ernst, Y.-H. Lai, W. Schnelle, Y.-H. Chu,Y. Sun, et al. , Thickness dependence of the anomaloushall effect in thin films of the topological semimetal co 2mnga, Physical Review B , 054422 (2019).[38] G.-H. Park, H. Reichlova, R. Schlitz, M. Lammel,A. Markou, P. Swekis, P. Ritzinger, D. Kriegner, J. Noky,J. Gayles, et al. , Thickness dependence of the anomalousnernst effect and the mott relation of weyl semimetal co 2mnga thin films, Physical Review B , 060406 (2020).[39] J. K¨ubler and C. Felser, Weyl points in the ferromagneticheusler compound co2mnal, EPL (Europhysics Letters) , 47005 (2016).[40] G. Chang, S.-Y. Xu, H. Zheng, B. Singh, C.-H. Hsu,G. Bian, N. Alidoust, I. Belopolski, D. S. Sanchez,S. Zhang, et al. , Room-temperature magnetic topologi-cal weyl Fermion and nodal line semimetal states in half-metallic heusler co 2 tix (x= si, ge, or sn), Scientific re-ports , 38839 (2016).[41] Z. Wang, M. Vergniory, S. Kushwaha, M. Hirschberger,E. Chulkov, A. Ernst, N. P. Ong, R. J. Cava, and B. A.Bernevig, Time-reversal-breaking weyl Fermions in mag-netic heusler alloys, Physical review letters , 236401(2016).[42] S. Chadov, S.-C. Wu, C. Felser, and I. Galanakis, Stabil-ity of weyl points in magnetic half-metallic heusler com-pounds, Physical Review B , 024435 (2017).[43] S. K. Kushwaha, Z. Wang, T. Kong, and R. J. Cava,Magnetic and electronic properties of the cu-substitutedweyl semimetal candidate zrco2sn, Journal of Physics:Condensed Matter , 075701 (2018).[44] M. Yang, G. Gu, C. Yi, D. Yan, Y. Li, and Y. Shi, Mag-netic and transport properties of zr1- x nb x co2sn, Jour-nal of Physics: Condensed Matter , 275702 (2019).[45] J. Barth, G. H. Fecher, B. Balke, S. Ouardi, T. Graf,C. Felser, A. Shkabko, A. Weidenkaff, P. Klaer, H. J.Elmers, et al. , Itinerant half-metallic ferromagnets co 2ti z (z= si, ge, sn): Ab initio calculations and measure-ment of the electronic structure and transport properties,Physical Review B , 064404 (2010).[46] J. Barth, G. H. Fecher, B. Balke, T. Graf, A. Shkabko,A. Weidenkaff, P. Klaer, M. Kallmayer, H.-J. Elmers,H. Yoshikawa, et al. , Anomalous transport propertiesof the half-metallic ferromagnets co2tisi, co2tige andco2tisn, Philosophical Transactions of the Royal Soci-ety A: Mathematical, Physical and Engineering Sciences , 3588 (2011).[47] R. Ooka, I. Shigeta, Y. Sukino, Y. Fujimoto, R. Y.Umetsu, Y. Miura, A. Nomura, K. Yubuta, T. Yamauchi,T. Kanomata, et al. , Magnetization and spin polarizationof heusler alloys co { } tisn and co { } tiga { . } sn { . } , IEEE Magnetics Letters , 1 (2016).[48] L. Bainsla and K. Suresh, Spin polarization studies inhalf-metallic co2tix (x= ge and sn) heusler alloys, Cur-rent Applied Physics , 68 (2016).[49] I. Shigeta, Y. Fujimoto, R. Ooka, Y. Nishisako, M. Tsu-jikawa, R. Y. Umetsu, A. Nomura, K. Yubuta, Y. Miura, T. Kanomata, et al. , Pressure effect on the magneticproperties of the half-metallic heusler alloy co 2 tisn,Physical Review B , 104414 (2018).[50] J. Hu, J. Niu, B. Ernst, S. Tu, A. Hamzi´c, C. Liu,Y. Zhang, X. Wu, C. Felser, and H. Yu, Unconven-tional spin-dependent thermopower in epitaxial co2ti0.6v0. 4sn0. 75 heusler film, Solid State Communications ,113661 (2019).[51] J. Hu, B. Ernst, S. Tu, M. Kuveˇzdi´c, A. Hamzi´c, E. Tafra,M. Basleti´c, Y. Zhang, A. Markou, C. Felser, et al. ,Anomalous hall and nernst effects in co 2 ti sn and co2 ti 0.6 v 0.4 sn heusler thin films, Physical Review Ap-plied , 044037 (2018).[52] R. Dunlap and G. Stroink, Conduction electron contribu-tions to the sn hyperfine field in the heusler alloy co2ti1-xvxsn, Journal of Applied Physics , 8210 (1982).[53] W. Pendl Jr, R. Saxena, A. Carbonari, J. Mestnik Filho,and J. Schaff, Investigation of the magnetic hyperfinefield at the y site in the heusler alloys (y= ti, v, nb,cr; z= al, sn), Journal of Physics: Condensed Matter ,11317 (1996).[54] H. Ebert et al. , The munich spr-kkr package, version 6.3,URL:¡ http://ebert. cup. uni-muenchen. de/SPRKKR(2005).[55] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalizedgradient approximation made simple, Physical review let-ters , 3865 (1996).[56] H. Ebert, D. Koedderitzsch, and J. Minar, Calculatingcondensed matter properties using the kkr-green’s func-tion method—recent developments and applications, Re-ports on Progress in Physics , 096501 (2011).[57] P. Soven, Coherent-potential model of substitutional dis-ordered alloys, Physical Review , 809 (1967).[58] P. Soven, Contribution to the theory of disordered alloys,Physical Review , 1136 (1969).[59] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ-cioni, I. Dabo, et al. , Quantum espresso: a modularand open-source software project for quantum simula-tions of materials, Journal of physics: Condensed matter , 395502 (2009).[60] N. Marzari and D. Vanderbilt, Maximally localized gen-eralized wannier functions for composite energy bands,Physical review B , 12847 (1997).[61] I. Souza, N. Marzari, and D. Vanderbilt, Maximally local-ized wannier functions for entangled energy bands, Phys-ical Review B , 035109 (2001).[62] G. Pizzi, V. Vitale, R. Arita, S. Bl¨ugel, F. Freimuth,G. G´eranton, M. Gibertini, D. Gresch, C. Johnson,T. Koretsune, et al. , Wannier90 as a community code:new features and applications, Journal of Physics: Con-densed Matter , 165902 (2020).[63] D. Hamann, Optimized norm-conserving vanderbilt pseu-dopotentials, Physical Review B , 085117 (2013).[64] F. Birch, Finite elastic strain of cubic crystals, Physicalreview , 809 (1947).[65] P. Van Engen, K. Buschow, and M. Erman, Magneticproperties and magneto-optical spectroscopy of heusleralloys based on transition metals and sn, Journal of Mag-netism and Magnetic Materials , 374 (1983).[66] A. Carbonari, R. Saxena, W. Pendl Jr, J. Mestnik Filho,R. Attili, M. Olzon-Dionysio, and S. De Souza, Magnetichyperfine field in the heusler alloys co2yz (y= v, nb, ta,cr; z= al, ga), Journal of magnetism and magnetic ma- terials , 313 (1996).[67] V. Popescu and A. Zunger, Effective band structureof random alloys, Physical review letters , 236403(2010).[68] V. Popescu and A. Zunger, Extracting e versus k effectiveband structure from supercell calculations on alloys andimpurities, Physical Review B , 085201 (2012).[69] S. Lee, T. D. Lee, P. Blaha, and K. Schwarz, t), Journalof applied physics , 10C307 (2005).[70] M. Hickey, A. Husmann, S. Holmes, and G. Jones, Fermisurfaces and electronic structure of the heusler alloyco2tisn, Journal of Physics: Condensed Matter , 2897(2006).[71] H. C. Kandpal, G. H. Fecher, and C. Felser, Calculatedelectronic and magnetic properties of the half-metallic,transition metal based heusler compounds, Journal ofPhysics D: Applied Physics , 1507 (2007).[72] A. Aguayo and G. Murrieta, Density functional studyof the half-metallic ferromagnetism in co-based heusleralloys co2msn (m= ti, zr, hf) using lsda and gga, Journalof Magnetism and Magnetic Materials , 3013 (2011).[73] J. K¨ubler and C. Felser, Berry curvature and the anoma-lous hall effect in heusler compounds, Physical Review B , 012405 (2012).[74] I. Galanakis, P. Dederichs, and N. Papanikolaou, Slater-pauling behavior and origin of the half-metallicity of thefull-heusler alloys, Physical Review B , 174429 (2002).[75] Vineeta shukla, Shiv Om kumar, Destruction of Half-metallicity in Co VSn Heusler alloy Due to X-Y Swap-ping disorder, Journal of Superconductivity and novelmagnetism (2020).[76] Wang, Xinjie, Yates and Jonathan R, Ab initio cal-culation of the anomalous Hall conductivity by Wannierinterpolation, Physical Review B , 191158 (2006). Appendix A: Electronic structures of stoichiometriccompounds
First, we look at the band structure of Co TiSn (Fig.7(a)). When SOC is not considered, the compound pos-sesses three mirror symmetries, M x , M y , and M z (alongthe planes k x = 0, k y = 0, and k z = 0), and three C rotation axes k x , k y , and k z . Due to these symmetries,there exist nodal lines in the momentum space on each ofthe three planes, formed by the crossings of the spin-upstates of the Co-Ti hybridized 3 d electron bands. Thesecrossings can be seen in the band structures in Fig. 7along the Γ − X , Γ − K and Γ − W high-symmetry lines,circled in blue. The energy of these nodal lines oscillatesaround the Fermi energy, reaching a maximum energy of0.45 eV, along the Γ − K direction, and a minimum of-0.35 eV, along the Γ − X direction. The high-symmetrypoints and the directions followed in the band structuresare the same as illustrated in Fig. 2(b).When Ti is replaced by V, from Co TiSn to Co VSn,an additional 3 d electron is added. As a result, inCo VSn, one of the spin-up Co-V (previously Co-Ti) hy-bridized 3 d bands, which had higher energy in Co TiSn,now lies on the Fermi energy. The Fermi energy itself in-creases, and the other states below the Fermi level only FIG. 7: Spin-polarized band structures of Co TiSn andCo VSn with (a),(c) calculated lattice parameters and(b),(d) experimental lattice parameters. The black linesrepresent majority spin states, and the red linesrepresent minority spin states. The blue circles showthe majority band crossings that form nodal lines in themomentum space.shift rigidly maintaining the same band shape. There isno change in the shape of the spin-down states across allenergy levels, and the half-metallic character is retained.The minority spin band gap and total magnetic mo-ment per formula unit are 0.491 eV and 2.03 µ B , and0.546 eV and 3.01 µ B , for Co TiSn and Co VSn respec-tively, in good agreement with literature [45, 69–72]. Thedifference in the minority spin band gap, which arisesfrom the splitting of the anti-bonding Co 3 d orbitals[72, 74], is nearly 0.05 eV. This difference may be at-tributed to the difference in lattice parameters of the twocompounds [71].Fig. 7 (a) and (c) show the spin-polarised, non spin-orbit coupled band structures of Co TiSn and Co VSnobtained using the calculated lattice parameters. Non-SOC band structures provide an insight into the symme-try of the bands, and how the symmetry changes whenSOC is considered and Weyl nodes are formed. Throughthese structures we can also study how the hybridisa-tion of these bands changes on inclusion of spin-orbitcoupling. When SOC is considered, the mirror symme-tries are broken according to the direction of magneti-zation. Due to this, the nodal lines gap out results inWeyl points, and there is a large Berry curvature in thevicinity of Fermi level.The location of these nodes nat-urally depends on the direction of magnetization, sinceit determines which symmetries are broken. Wang etal. [41] performed a symmetry analysis for the Co XZ(X = IVB or VB; Z = IVA or IIIA) Heusler compoundsto determine high-symmetry lines and planes which maycontain Weyl nodes under different magnetization direc-1tions. With magnetization set in the [110] direction, theydetermined that Weyl nodes may be found along the [110]axis and on the xy -plane, and nodal lines may exist onthe ([1¯10], [001]) plane. When the magnetization is along [001], Weyl nodes may appear along the [001] axis, andon the xz -plane and yz -plane, and the nodal line on the xyxy